Question

Is Thevenin theorem true for linear or nonlinear?

Answer 1

Thevenin's theorem is applicable **only to linear circuits**. This means it is valid for circuits that consist of linear elements such as resistors, capacitors, inductors, linear dependent sources, and independent sources.

### Why is it limited to linear circuits?

- **Linearity** means that the relationship between voltage and current follows Ohm's Law or other linear relationships (e.g., V = IR, I = C(dV/dt), etc.), and the superposition principle holds. In linear circuits, the response of the circuit (voltage or current) is proportional to the input.
  
- **Nonlinear circuits**, on the other hand, include components like diodes, transistors, or any element where the relationship between voltage and current is nonlinear. In such cases, Thevenin's theorem does not apply directly because the equivalent resistance and voltage source concepts are not valid due to the nonlinear behavior of the elements.

For nonlinear circuits, other analysis methods, like numerical simulations or more specialized theorems, are typically required.

Answer 2

The Thevenin Theorem applies specifically to **linear** circuits. Let's break down why this is the case and what it means.

### Thevenin Theorem Overview

Thevenin's Theorem states that any linear, bilateral network of resistors and sources can be simplified to a single voltage source (the Thevenin equivalent voltage) in series with a single resistor (the Thevenin equivalent resistance) when viewed from two terminals. This simplification makes analyzing and solving circuits much easier, especially in complex networks.

### Linearity Requirement

For the Thevenin Theorem to be valid, the circuit must be linear. Here's why linearity is crucial:

1. **Superposition Principle**: Thevenin’s Theorem relies on the superposition principle, which states that in a linear circuit, the response (voltage or current) caused by multiple sources is equal to the sum of the responses caused by each source acting independently. Nonlinear circuits do not follow this principle, as the response to multiple sources in such circuits is not simply additive.

2. **Constant Proportionality**: In linear circuits, the relationship between voltage and current (Ohm's Law) is proportional and constant. For example, in resistive networks, Ohm's Law (\(V = IR\)) holds true, and this proportionality is key to finding the Thevenin equivalent. Nonlinear components, such as diodes or transistors in their active regions, do not have a constant proportional relationship, which complicates the simplification process.

3. **Superposition of Impedances**: In linear circuits, impedance (resistance, in the case of resistors) remains consistent regardless of the superimposed signals. In nonlinear circuits, impedance can vary with the signal level, making the Thevenin equivalent analysis invalid.

### Nonlinear Circuits

For nonlinear circuits, Thevenin’s Theorem doesn’t directly apply because:

- **Variable Relationships**: Nonlinear elements, like diodes or transistors, have relationships between voltage and current that are not linear. For example, the current through a diode is an exponential function of the voltage.

- **Complex Behavior**: The behavior of nonlinear components can be highly complex and dependent on the signal's magnitude, frequency, and waveform, making a simple linear equivalent impossible in many cases.

### Approximate Methods for Nonlinear Circuits

In practical scenarios, engineers sometimes use linear approximations of nonlinear circuits around specific operating points, but these are approximations rather than exact solutions. For instance:

- **Small-Signal Analysis**: In analog electronics, nonlinear devices are often analyzed using small-signal models around a bias point. These models are linear approximations that work for small deviations around the operating point.

- **Piecewise Linear Models**: Some nonlinear devices are approximated with piecewise linear models, which can simplify analysis but only within certain ranges of operation.

### Summary

Thevenin's Theorem is strictly valid for linear circuits due to its reliance on the superposition principle and constant proportional relationships. For nonlinear circuits, alternative analysis methods must be used, often involving linear approximations or numerical techniques, to handle their more complex behaviors.